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We know generators and relations for $\mathfrak{sl}_2$: $e, f, h, [e, f]=h, [h,e]=2e, [h,f]=-2f$. Generators and relations for $U_q(\mathfrak{sl}_2)$ are $e, f, k=q^h$, $kek^{-1}=q^2e, kfk^{-1}=q^{-2}f, [e, f]=\frac{k-k^{-1}}{q-q^{-1}}$.

What are generators and relations for $\hat{\mathfrak{sl}}_2$ and $U_q(\hat{\mathfrak{sl}}_2)$? Thank you very much.

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